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\title{多元统计分析练习5.1-5.2}
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\date{2024 年 5 月 14 日}
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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
判别分类要解决的问题是，在已知历史上用某些方法已把研究对象分成若干组的情况下，来判定新的观测样品应归属的组别。举例说明判别分析要解决的问题。

\vspace{0.2cm}

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\item  %Problem 02
设两组 $\pi_1$ 和 $\pi_2$ 的均值分别为 $\mu_1$ 和 $\mu_2$, 协方差矩阵分别为 $\Sigma_1$ 和 $\Sigma_2$. 
设 $\Sigma_1=\Sigma_2=\Sigma$. 设 $x$ 是一个新的样品，使用距离判别法判断它来自哪一组。
\begin{enumerate}
\item  写出平方马氏距离的计算公式和判别规则。
\item  设两组均为正态总体，写出误判概率 $P(2\mid 1)$ 和 $P(1\mid 2)$ 的计算公式。
\end{enumerate}

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\item  %Problem 03
例子5.2.1. 设两组 $\pi_1$ 和 $\pi_2$ 的分布分别为 $N(\mu_1,\sigma^2)$ 和 $N(\mu_2,\sigma^2)$. 
设参数 $\mu_1,\mu_2,\sigma^2$ 均已知。使用距离判别法，导出判别规则和误判概率。

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\item  %Problem 04
设有来自组 $\pi_1$ 的样本 $x_{11}, x_{12}, \cdots, x_{1n_1}$ 和来自组 $\pi_2$ 的样本 $x_{21}, x_{22}, \cdots, x_{2n_2}$. 
\begin{enumerate}
\item  写出距离判别法的判别规则。
\item  设两个都是正态组，写出误判概率的计算公式。
\end{enumerate}

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\item  %Problem 05
设两组不能假设为正态总体，写出误判概率的计算公式。
\begin{enumerate}
\item  使用回代法。
\item  使用划分样本的方法。
\item  使用交叉验证法。
\end{enumerate}

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\item  %Problem 06
设两组 $\pi_1$ 和 $\pi_2$ 的均值分别为 $\mu_1$ 和 $\mu_2$, 协方差矩阵分别为 $\Sigma_1$ 和 $\Sigma_2$. 
设 $\Sigma_1\neq \Sigma_2$. 设 $x$ 是一个新的样品，写出距离判别法的判别规则。

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\item  %Problem 07
例子5.2.2. 设两组 $\pi_1$ 和 $\pi_2$ 的分布分别为 $N(\mu_1,\sigma_1^2)$ 和 $N(\mu_2,\sigma_2^2)$. 
设参数 $\mu_1,\mu_2,\sigma_1^2,\sigma_2^2$ 均已知。
使用距离判别法，导出判别规则和误判概率。


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\item  %Problem 08
设有 $k$ 个组 $\pi_1,\pi_2,\cdots,\pi_k$, 均值分别为 $\mu_1,\mu_2,\cdots,\mu_k$, 协方差矩阵分别为 $\Sigma_1,\Sigma_2, \cdots,\Sigma_k$.  写出距离判别法的判别规则。
\begin{enumerate}
\item  设均值和协方差矩阵都已知。
\item  设均值和协方差矩阵都未知。
\end{enumerate}

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\item  %Problem 09
例子5.2.3. 对破产的企业收集它们在破产前两年的年度财务数据，同时对财务良好的企业也收集同一时期的数据。
数据涉及四个变量：$x_1=$现金流量/总债务，$x_2=$净收入/总资产，$x_3=$流动资产/流动债务，$x_4=$流动资产/净销售额。数据参见表格5.2.1. I组为破产企业，II组为非破产企业。
现有某未判企业 $x=(-0.16, -0.10, 1.45, 0.51)'$, 使用线性判别规则进行判别。
分别使用回代法和交叉验证法计算误判概率。

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\item  %Problem 10
设对来自组 $\pi_1$ 和组 $\pi_2$ 的两个样本有 
$$
\bar{x}_1=\begin{pmatrix} 4 \\ 2 \end{pmatrix}, \,\,\, 
\bar{x}_2=\begin{pmatrix} 3 \\ -1 \end{pmatrix}, \,\,\, 
S_p=\begin{pmatrix} 6.5 & 1.1 \\ 1.1 & 8.4 \end{pmatrix}, 
$$
设 $\Sigma_1=\Sigma_2$, 试给出距离判别规则，并将 $x_0=(2,1)'$ 分到组 $\pi_1$ 或 $\pi_2$. 

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\end{enumerate}


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\end{document}

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